Reducing access times for radiation treatment by aligning the doctor’s
schemes
Ingeborg A. Bikker
a,b,∗, Nikky Kortbeek
a,b, Rob M. van Os
c, Richard J. Boucherie
a aCentre for Healthcare Operations Improvement and Research (CHOIR), University of Twente, Drienerlolaan 5, 7500 AE Enschede, The Netherlands bDepartment of Quality and Process Innovation, Academic Medical Center, Meibergdreef 9, 1105 AZ Amsterdam, The NetherlandscRadiotherapy Department, Academic Medical Center, Meibergdreef 9, 1105 AZ Amsterdam, The Netherlands
a r t i c l e i n f o Article history:
Received 27 November 2014 Accepted 30 June 2015 Available online 14 August 2015
Keywords: Radiotherapy Capacity allocation Linear programming Simulation
a b s t r a c t
Around 40% of cured cancer patients in the European Union are treated with radiotherapy [2]. Delays in cancer treatment are associated with psychological distress and decreased cancer control. To this end, in the Netherlands standards for the access time for radiation treatment are set, which are currently not met in many Dutch oncological centers. The radiotherapy care process (i.e., preparation and treatment) consists of several consecutive stages, possibly related via time constraints. Inadequate capacity allocation may cause large delays, for example due to the capacity allocation of different stages not being aligned, or due to inadequate time division of single resources over different activities. The objective of this study is to increase compliance to access time standards without extending resource capacities, by developing a methodology for optimizing resource capacity allocation in the radiotherapy care process.
For radiotherapy, time division of resources over different activities particularly applies to the doctors, who carry out consultations and scan contouring. Time slots for these activities are typically set for each doctor in a cyclic weekly scheme. We develop an integer linear programming (ILP) model to design a weekly doctors’ scheme that minimizes the expected access times of all patient types in the care process and that matches the number of consultation time slots with demand. In several experiments, the quality of the resulting doctors’ schemes is studied via a discrete event simulation model by evaluating the consequences of the schemes in a stochastic environment. Results from a case study in the Academic Medical Center (AMC) in Amsterdam show that the implementation of these schemes may result in a considerable access time reduction. The designed doctor’s schemes are being evaluated for implementation in the AMC.
© 2015 Elsevier Ltd. All rights reserved.
1. Introduction
This paper presents a methodology for designing a weekly cyclic doctors’ scheme for radiotherapy care, providing time slots for different activities in the care process for each doctor. This scheme is designed in such a way that compliance with access time standards (concerning the number of calendar days between referral and start of the treatment) can be increased. The methodology is applied to a case study in the Academic Medical Center Amsterdam (AMC), a Dutch university hospital.
∗Corresponding author at: Center for Healthcare Operations Improvement and
Research (CHOIR), University of Twente, Drienerlolaan 5, 7500 AE Enschede, The Netherlands.
E-mail address:i.a.bikker@utwente.nl(I.A. Bikker).
Radiotherapy is one of the most commonly used cancer therapies worldwide, along with surgery and chemotherapy [1,2]. It is based on the medical use of ionizing radiation to control or kill malignant cells. Radiotherapy may be curative depending on the type and stage of cancer, or it is used for palliative purposes in the sense of pain relief. A majority of the radiotherapy patients are treated with external beam radiation produced by a linear accelerator (linac). Before this treatment can start, depending on the patient type several preparation stages need to be performed (amongst others consultation, CT scan, scan contouring). Time constraints are involved between some of the stages. In most radiotherapy facilities, consultations and scan contouring take place according to weekly cyclic doctor’s schemes.
Delays in treatment are associated with psychological dis-tress [3] and decreased cancer control [4], therefore the radiation sessions should preferably start as soon as possible after the re-ferral. To this end, standards for the access times for radiation
http://dx.doi.org/10.1016/j.orhc.2015.06.005
Fig. 1. Example of a doctor’s scheme and a patient time schedule.
treatment are set in many countries [4,5]. In the Netherlands, the prevailing standards are currently not met in many oncological centers. Several factors may delay the start of the treatment. Note that we focus on the access time between referral and start of the treatment, measured in number of calendar days; the wait-ing time for the appointments (time spent in the waitwait-ing room) is not the main focus of our work. Different patient types require dif-ferent appointments with possible time constraints between ap-pointments. Various care professionals are involved in the care process, who all have their own tasks and schedules. Doctors di-vide their time over consultation time slots, scan contouring and other activities, and some stages have limited availability in the week. Variability in patient arrivals further complicates the pro-cess. For the Radiotherapy department in the AMC, we considered historical data and observed that delays occur, even when the re-source capacity at separate stages in the care process is sufficient to avoid congestion. Based on historical data and insights from lit-erature, we identify the following factors that may cause delays in the AMC radiotherapy care process: (1) the opening hours or time slots allocated to stages are not aligned to the capacity allocation in preceding and consecutive stages in the care process, (2) the daily number of consultation time slots is not aligned to the demand on daily level, and (3) the consultation slots for different patient types are not distributed equally over the week.
As an illustration for the planning of a radiotherapy treatment, we consider a doctor with two consultation time slots on Thurs-day and a contouring time slot on MonThurs-day. A patient who is re-ferred on Monday might have a consultation on Thursday. If a CT scan is made on Tuesday, the first possibility for scan contour-ing (i.e., markcontour-ing the edges of the tumor) is on Monday. When five working days are incorporated for treatment planning, the linac sessions may start on the next Monday. This results in an ac-cess time of 21 calendar days (including weekends). A graphical overview of the example is displayed inFig. 1. Merely, interchang-ing the doctors’ consultation and contourinterchang-ing time slots would, for this patient, have reduced the access time by four days (20%).
We develop a model for the capacity allocation of doctors to their multiple activities, aligned with the demand and with the capacity allocation in consecutive stages. The objective is to in-crease compliance to the access time standards while efficient resource utilization is maintained. The method is based on an in-teger linear program (ILP) and the resulting schemes are evaluated by discrete event optimization. The potential effectiveness of this methodology is demonstrated by its application to a case study in the AMC. To incorporate the particular characteristics of the AMC, the method is developed in close cooperation with radiotherapy care experts. The results of the AMC case demonstrate the appli-cation of our models in radiotherapy care to be very promising. By implementing the methodology, patient treatment can start earlier while using the same resource capacity.
This paper is organized as follows. Section 2 provides an overview of the related literature. In Section3, the radiotherapy
care process and the specific characteristics of the process in the AMC are described. In Section4, we introduce our optimization model. Section5describes the results of the case study, followed by the conclusion and discussion in Section6.
2. Related literature
The methodology presented in this paper comprises the design of a blueprint doctors’ scheme for the radiotherapy care process, in which many consecutive stages and multiple resources are involved that all have their own constraints. To place our contribution in the context of previous work, we describe the subdivision of capacity planning decisions into different levels, followed by an overview of logistics research performed in the area of radiotherapy and an overview of work on appointment schemes in other healthcare areas.
Planning and control in healthcare has received increased attention over the last ten years, both in literature and in practice. Operations Research (OR) plays an important role in addressing logistics healthcare challenges, especially related to resource capacity planning and control [6]. A commonly applied subdivision of capacity planning and control decisions is the division in three hierarchical levels: the strategic, tactical and operational level [7]. The strategic level focuses on long-term planning decisions such as how many machines or equipment to buy. The operational level considers short-term decisions such as appointment scheduling (determining a consultation time and date for a specific patient). The tactical level connects these levels, covering mid-term decision problems such as how to allocate resource capacity to patient groups or different processes, for example in the form of designing blueprint appointment schemes that may be used to provide a specific time and date for patient consultation. Hulshof et al. [8] provide a comprehensive overview of capacity planning and control decisions in all of these hierarchical levels. They observe that the majority of OR in healthcare contributions focused on a single facility or department, while an integrated approach concerning multiple departments or stages in the care process may lead to better results [9].
Literature on capacity planning and control in radiotherapy has mainly focused on the operational planning level and on separate parts of the care process, such as appointment scheduling on linacs [10–16]. However, some strategic and tactical issues have been addressed. On the strategic level, Thomas [17] used discrete event simulation to evaluate the percentage of spare capacity that should be reserved to keep access times to treatment short. On the tactical level, Pérez Rivera developed guidelines to assign patients to linacs based on their tumor type and linac characterizations [18]. Joustra et al. [19] address the tactical question whether to pool (sub)acute and regular patients into different types for scheduling a consultation, and tactical guidelines to reduce supply variability are studied by Joustra et al. [20]. Only few studies consider the entire radiotherapy care process [21–23], but to the best of our
Fig. 2. General care process for regular and subacute patients treated on a linac.
knowledge there have not been studies on blueprint appointment schemes incorporating the entire radiotherapy care process.
The design of blueprint appointment schemes has been con-sidered in other healthcare planning problems [24,25]. Most stud-ies focus on single appointments or services from single resources [26,27]. Some studies discuss combination appointments with many different resources and precedence constraints in a multi-disciplinary setting. Braaksma et al. [28] present a methodology to plan appointment series for rehabilitation outpatients, based on an ILP formulation. The combination appointment scheduling in the ‘Children’s Muscle Center Amsterdam’ [29] and the schedul-ing of multi-step sequential procedures in nuclear medicine [30] are both examples of multidisciplinary scheduling on one day, re-quiring many resources and interrelated appointment constraints. In these articles, the access time is considered as a key perfor-mance indicator. Care planning from multiple health services and scheduling in highly constrained situations are identified as open challenges in the comprehensive review on appointment schemes in outpatient departments by Gupta and Denton [25].
Our contribution will be on the tactical level. We present a methodology for designing a blueprint appointment scheme for radiotherapy care, considering the numerous constraints and objectives that apply to radiotherapy care planning, taking into account the relation between different stages such that compliance to the access time standards can be increased.
3. Radiotherapy care process
This section presents the different stages of the radiotherapy care process and the (time) constraints between the stages. In the context of the AMC case, we state the Dutch access time standards and we describe the specific characteristics, scheduling procedures and planning rules of the Radiotherapy department in the AMC.
Before the radiation treatment can start, several preparation stages need to be performed. These stages are graphically shown
inFig. 2. At the referral, a consultation is scheduled with a doctor
specialized in the patient’s tumor type. In some cases, additional appointments are needed, but in general the next stage is an appointment on the CT simulator (a CT scanner that is also used for body position verification). When the CT scan is made, a doctor contours the tumor on the scan. The contouring is usually done by the doctor to whom the patient is assigned. This contoured scan is used as a basis for the treatment plan (a detailed description of the dose of radiation and the angles of the radiation beams), made by a radiation technologist. The radiation sessions are executed on a linac by radiation technologists on consecutive days, with exception of the weekends. The number of sessions usually varies between one and forty, depending on the size and type of cancer and the patient’s condition [31].
Some patients need other appointments additional to the standard care process. For example, prostate and digestive patients get beads implemented in the body for position verification of the relevant organs on the CT scan. After implementation of these beads, the body has to recover for several days before an accurate CT scan can be made. For some patient types it is necessary to
make different scans or visualizations in addition to, or instead of the CT scan: a PET-CT scan, 4D-CT scan or cystoscopy. Further, extra devices may have to be molded before the treatment can start. A mold can be made to fix a specific body part, or to control the shape of the linac beams. In both cases, the mold has to be tested on a linac in the patient’s presence. Radiotherapy treatment can be combined with chemotherapy, surgery, brachytherapy (implementing a radioactive source in the patient’s body) or hyperthermia (heating a part of the patient’s body). Hyperthermia sessions have to be scheduled in between the linac sessions and usually take place once a week; the other treatments normally have to be scheduled within a time frame before or after the linac sessions, so coordination with other departments is required.
The following access time standards, process characteristics and planning rules specifically apply to the radiotherapy process in the AMC. Dutch access time standards state that 80% of subacute (palliative) patients have to start sessions within seven calendar days after referral and 100% have to start within ten calendar days. For regular (curative) patients the standard is that 80% have to start sessions within 21 calendar days after referral and 100% within 28 calendar days.
The Radiotherapy department of the AMC employs twenty doc-tors and physician assistants and about 50 radiation technologists, who jointly carried out 29,940 radiation sessions on 1981 patients in 2012. The department operates on two treatment locations: the main location in the AMC in Amsterdam and a smaller satellite lo-cation in the Flevo hospital in Almere. Doctors spend their working time divided between these locations. The consultations, CT simu-lator appointments and radiation sessions can take place at both locations. Scan contouring can be done on both locations for all patients, independent of where the patient is treated. Additional stages (PET-CT, 4D-CT or cystoscopy) and combination treatments are only performed at the main location.
Consultations and scan contouring take place according to a cyclic weekly doctor’s scheme. The consultation is scheduled at the moment of referral, in the first available consultation slot of a doctor specialized in the patient’s tumor type. In the AMC, the patient’s treatment proposal is discussed in the so-called doctors’ meeting on the next working day after the consultation. The remaining appointments are scheduled after the doctors’ meeting. To allow enough time for development of the treatment plan and entering the treatment details, a scheduling guideline is asserted for the time between the scan contouring and the start of the sessions. For patients treated in Amsterdam, this time is set to five working days for regular patients and two for subacute patients. For patients treated in the Flevo hospital, one extra day is added for transportation of the required paper patient records.
Regular patients do not start their treatment on Friday, because receiving one session of radiation before two days ‘off’ is considered to be ineffective for the treatment. Instead of a standard consultation, subacute patients can be scheduled for a one stop shop consultation, where the stages from the consultation until the first radiation session take place on one day and the doctors’ meetings are skipped.
4. Approach
This section presents a methodology to develop a doctors’ scheme that provides the time slots for consultations and scan contouring for each doctor, with the purpose to increase the compliance to the access time standards. The length of the access time of a patient depends on several factors, which are listed below. For organizational reasons, doctor’s schemes need to be weekly cyclic. As a consequence, these schemes are determined before dynamical factors such as the exact number and type of referred patients in a given week are known. For this reason, we distinguish static factors that are known at the moment the scheme is created, and dynamical factors that are unknown at the moment the scheme is constructed, representing variations in demand and supply. The static factors are:
•
the doctors’ skills for treating specific patient types;•
the doctors’ availability for consultations and contouring (e.g., doctors might have part time days, educational obligations and multidisciplinary meetings);•
the (predetermined) number of consultation time slots and contouring time slots to be scheduled for each doctor and the locations where the doctors are stationed;•
the medically prescribed care pathways and the guidelines for patient routing (that is, determining which stages to pass through and the time constraints);•
the days on which different stages are carried out (e.g., gold beads implementation only takes place one morning a week);•
the appointment scheduling guidelines used by the department (e.g., scheduling five working days between contouring and start of the treatment).The dynamical factors are:
•
the stochastic arrival pattern of patients at different referral days on both locations;•
the number of consultation time slots that vary due to temporary absence of doctors for conferences, holidays and illness; doctors in training change during the year due to internships in several hospitals.To capture the complexity of the radiotherapy care process including all factors listed above, we present a combined approach of ILP and discrete event simulation that combines the strengths of both techniques. ILP is a suitable technique for scheduling problems, but is not suitable for dynamic optimization. Computer simulation can handle dynamical factors and is often used for evaluating complex, detailed systems, but it is not suitable for optimization. Combining the two enables us to evaluate an optimal solution in a realistic, dynamical environment. Such a combined approach is a common and suitable method for optimization problems in industry and healthcare [32]. Our approach consists of two parts: first, we use the static parameters to optimize the doctors’ scheme (ILP model), and then the scheme is evaluated in a stochastic environment that incorporates the dynamical factors (simulation).
The doctors’ scheme is created as follows. For a certain doctors’ scheme, we can calculate the access time of one arriving patient in a system without already scheduled appointments. The access time of a patient under these conditions is a ‘lower bound’ of the access time, since it is the shortest access time a patient could experience when this specific doctors’ scheme is used. This calculation is carried out for each patient type, arriving on each referral day on each location. Merging these lower bounds, we derive a weighted lower bound for all patients. Then, we consider all doctors’ schemes and select the schemes for which the weighted lower bound is minimal. Given the schemes with a minimal lower bound, as a secondary objective, we select the scheme that balances the daily
demand and supply of consultation time slots. Finally, the resulting scheme is evaluated in a discrete event simulation where also the dynamical parameters are taken into account, such that the consequences of the scheme and of the variable patient arrivals on the access times can be analyzed.
For tractability, we cluster all possible diagnoses into patient types. For each patient type there is a protocol prescribing the stages that need to be performed and the time constraints for scheduling the appointments. We group patients together based on focus area and similar protocols, including stages the patient has to pass through and the time constraints between the stages.
4.1. ILP model to design the doctors’ scheme
We model the construction of the weekly doctors’ scheme for consultation time slots and scan contouring as an ILP. To do so, we divide a week in time slots of equal length. The decisions to be made are the time slots for consultations and contouring for each doctor on both treatment locations. Thus, the decision variables are:
Xadl
=
1 if doctor a has a consultation time in time slot don location l 0 otherwise
,
Yadl
=
1 if doctor a has a contouring task in time slot don location l 0 otherwise
.
For each patient type, referral day and treatment location, the ‘lower bound’ of the access time is determined by calculating the access time in a system without already scheduled appointments. In order to do this, it is necessary to know to which doctor the patient is allocated and on which day the patient could have his/her appointments. The formal description of these variables and the mathematical formulation of the model can be found inTable 7
in theAppendix. The constraints and objectives are summarized below. ‘Patient’ refers to a patient of type p, referred on day r on location l.
Constraints. We distinguish the following constraints:
Basic constraints. A patient requires each appointment at most
once; a doctor can only do one activity at a time at one location, and only if he/she is available for outpatient care.
Constraints on the scheme construction. For each doctor, there
is a prescribed number of consultation time slots and contouring time slots per week on each location. The number of simultaneous consultation and contouring time slots is restricted to the facility’s capacity (consultation rooms, computer software licenses).
Constraints for patient–doctor allocation. Patients are allocated
to one doctor who performs the consultation as well as the contouring. The doctor must have the required specialization.
Constraints for patient-appointment allocation. For each patient
type, it is prescribed which appointments should take place and which precedence relations are involved. Some stages have to be performed before others and time restrictions apply between some appointments. Some patient types have additional restrictions for days on which treatment can start. Resources, and therefore stages, have limited availability.
Constraints for variable definition. These constraints are used for
the mathematical formulation. They denote whether a patient has an appointment on a certain day or not, the access time of a patient and the difference in daily supply and demand.
Objectives. The following two objectives are formulated:
Minimize the lower bounds of the access time. For each patient
type, referral day and treatment location, the lower bound of the access time for one patient is determined. These lower bounds are weighted by the occurrence rate of patient types for each referral
Fig. 3. Conceptual model of the radiotherapy care process.
day on each location. An extra weight factor for patient types is incorporated, since it might be possible that short access times for certain patient types (for example: subacute patients) are more important to the decision maker than short access times for other (for example: regular) patient types.
Minimize the difference between daily supply and demand.
A secondary objective is added to align the daily number of consultation time slots with the average daily patient arrivals, since matching supply and demand on a day-to-day level is valuable in the practical system where queueing effects apply. In the care process, the first possibility for a consultation is the first working day after the arrival. Therefore, we minimize the difference between the average number of arriving patients on a day and the number of consultation time slots on the consecutive working day, for each location.
In the objective function of the ILP model (see Appendix), the two objectives are weighted by factors
α
andβ
respectively. By setting the weight factors, the relative importance of each objective can be specified. The first objective determines the optimal moments for consultations and contouring, in relation to all other stages of the process and all other patient types, while the second objective only matches the total daily demand and supply of consultation time slots. Since we are interested in reducing the entire access time, we assign a higher weight to the first objective than to the second, in such a way that the second objective is used for fine tuning only and does not influence the outcome of the first objective.4.2. Simulation to evaluate effects of the doctors’ scheme
To evaluate the consequences of a doctors’ scheme in a stochastic environment, a discrete event simulation model is developed that captures the entire care process and takes queueing effects and variability into account.
A simplified version of the conceptual model is displayed in
Fig. 3. In our discrete event simulation model, the routing of a
patient is determined by his patient type, treatment location, type of consultation and the doctor to whom the patient is assigned. The patient type, location and type of consultation are randomly assigned at the moment of referral according to occurrence percentages. Patients that need an additional stage that can only be performed at a specific location are assigned to that location. Once the patient type, location and type of consultation are known,
a patient is assigned to the first available doctor in the associated location that can treat his patient type. A doctor is available if he/she is having an idle consultation time slot (according to the doctors’ weekly scheme). The consultations and contouring take place according to the doctors’ scheme. The treatment takes place on the first available linac (standard care process) or in the first available one stop shop time slot. When a patient arrives at a stage (denoted by a block inFig. 3), he/she is placed in a queue and served according to the ‘first come, first served’ policy. Patient arrivals take place according to a Poisson process, with service parameters that may differ per day of the week. Servers can handle patients during opening hours or time slots as prescribed by the doctors’ scheme. During the year, a certain percentage of the consultation and contouring time slots of a doctor are canceled due to holidays and conferences. On public holidays, all servers are unavailable.
The output data of the discrete event simulation model consists of the access times and time per stage, specified per location, per priority and per patient type (the average and 80- and 100-percentiles) that are measured retrospectively based on patient’s realized access time in the simulation model.
5. Experiments
This section presents the results of several experiments that are carried out for the AMC case. First, the experiments are described and the used parameter values are given, followed by the results of the experiments of the case study.
5.1. Experimental setup
In cooperation with the Radiotherapy department of the AMC, we decided which interventions to investigate and which experiments to carry out. The experiments are described below. Each experiment consists of the construction of a doctors’ scheme by the ILP model, and evaluation of the scheme in the simulation model. For reference, we also evaluated a doctors’ scheme that is currently used in the department.
•
Original: the current situation in the Radiotherapy department of the AMC with a currently used doctors’ scheme.•
Experiment 1: the current situation in the Radiotherapy department of the AMC is evaluated, using an optimized doctors’ scheme.•
Experiment 2: a change is considered in the patient types each doctor can treat: the two largest patient groups, breast and urology patients, can now be treated by all doctors. This experiment is set up because in the historical data, large entry times (number of days from referral to consultation) are recorded for these patient groups. This intervention is already daily practice in many hospitals. In this experiment, the doctors’ scheme is optimized with this new information.•
Experiment 3: in this experiment, the CT scan is scheduled on the same day as the consultation, for patient types that do not require additional stages. The doctors’ meeting is skipped. This experiment is set up because this intervention is currently considered in the AMC. In this experiment, the doctors’ scheme is optimized with this new information.•
Experiment 4: in this experiment, scan contouring is not restricted to time slots in the doctors’ scheme: it can be done on any moment within the doctor’s working hours. This has been daily practice in the AMC until several years ago, and is currently used in several other hospitals. We construct an optimized doctors’ scheme that does not contain contouring time slots.•
Experiment 5: the hypothetical situation is evaluated where the capacity of the CT simulators and linacs is extended to 24 h a day on working days. This experiment is set up to verify that the capacity of the CT simulators and the linacs is currently not the main bottleneck in the AMC. In this experiment, the original doctors’ scheme is used.•
Experiment 6: a combination of interventions as proposed in experiments 2, 3 and 4 is evaluated, to investigate if these logistical interventions would be enough to ensure that the access time standards are met.In order to carry out these experiments, the number of runs, the run length and the warm-up period need to be specified. The warm-up period and the run length are set based on an analysis of the performance indicators ‘access time’ and ‘time per stage’ for five test runs. The warm-up period is determined by applying Welch’s procedure [33] and is set to three months. Access times realized during the warm-up period are removed from the output data, as well as access times of patients that arrived in the last two months of each simulation run, since most of these patients do not finish the treatment before the end of the run which would bias the output data. The run length is set to four years, including the warm-up period and the last two months that are omitted. Based upon a desired half-width of 10% for the 95% confidence interval of the performance indicators ‘access times’ and ‘time per stage’, we used the sequential procedure as described in [33, pages 511–515] to set the number of replications at five per experiment. The ILP model is implemented in AIMMS 3.14 and is solved using CPLEX 12.6. The simulation model is implemented in Tecnomatix Plant Simulation 10.1 by Siemens PLM Software. For the experiments we used a 2.3 GHz Intel Core i5 HP Notebook with 8 GB RAM under a 64-bit version of Windows 7.
5.2. Parameter values
The input parameters for the models are based on the historical data of the AMC and on the characteristics of the AMC radiotherapy care process. The set values of the ILP model are displayed in
Table 1. The number of patient types and their routing information
is determined by grouping all possible diagnoses, based on the focus area, the stages to fulfill and time constraints between the stages. The number of time slots per day is set to two slots, because in practice, at most two activities per patient take place on a day. At the main location in Amsterdam, generally each doctor treats subacute patients and patients of two or three focus areas (for example, a doctor might treat digestive and urology patients). In
Table 1
Set values in the ILP model.
Description Number
Doctors 20
Patient types 18
Treatment locations 2
Stages 12
Time slots per day 2
Referral days per week 5
Time slots in the planning horizon
70
the Flevo hospital, all doctors treat all patient types. The routing information per patient type is displayed inTable 4. For the weight factors in the objective function, the order of magnitude of the objective values already provides the desired effect of a relative higher importance of the first objective. Therefore, the weight factors in the objective function are set to
α =
1 andβ =
1.For the simulation model, the distributions of the patient interarrival and service times are displayed in Table 2. The exponential and deterministic distribution parameters are based on historical data and the uniform distribution parameters are estimated by a doctor. The patient type distributions are given in
Table 3and the routing information per patient type is given in
Table 4. At the main location in Amsterdam, four linacs and one
CT simulator are available within certain opening hours. In the Flevo hospital, two linacs and one CT simulator are available within certain opening hours.
5.3. Results of the AMC case study
First, we comment on the results of the ILP model, followed by a comparison of the results of the ILP model and the simulation results. After that, the evaluation of the access times of the simulation model with respect to the access time standards is discussed, and the improvements are specified in more detail, by observing the times per stage and per patient type.
Table 5 presents the average lower bounds for all subacute
and regular patients, for the original scheme and for the schemes of experiments 1–6 obtained from our ILP model, as well as the average access times as realized in our simulation model. Note that the realized access times for subacute patients are in some cases lower than the lower bounds. This is due to the fact that the lower bounds for subacute patients are based on the procedures for patients that have a standard consultation type, whereas in the simulation model a percentage of the patients have a one stop shop appointment. It can be observed that both for regular and subacute patients, the average lower bound decreases substantially when the optimized doctors’ scheme is introduced (experiment 1). Further, we observe that the optimized scheme stays the same when each doctor can see breast and urology patients (experiment 2). For this reason, also the lower bounds for experiments 1 and 2 are equal. When the consultation and the CT simulator appointment can be scheduled on the same day (experiment 3), this influences the optimal number of days between a consultation time slot and a contouring time slot in the scheme, and lowers the values of the lower bounds as compared to experiment 1. For experiments 4 and 6, the lower bounds are almost equal to these of experiment 1. Experiment 5 uses the same scheme as the original situation. For illustration purposes, the original doctors’ scheme and the schemes resulting from experiments 1–2 and experiment 3 are displayed inFig. 4. Observe that the number of consultation time slots is more equally divided over the weekdays in the schemes created by the ILP model.
Table 5 also contains the access times from our simulation
model. Observe that the impact of introducing an optimized doctors’ scheme (experiments 1–4 and 6) is much larger than the
Consultation Deterministic 60 min According to doctors’ scheme
CT scan Deterministic 24 min Opening hours CT simulator (all working days)
4D-CT scan Deterministic 60 min Opening hours CT simulator (all working days)
PET-CT Deterministic 30 min Wednesday morning, 80 min
Gold beads Deterministic 30 min Monday morning, 150 min
Beads Deterministic 60 min Working days; 2–14 days waiting time (uniform)
Cystoscopy Deterministic 60 min Working days; 2–14 days waiting time (uniform)
Contouring Uniform 30–120 min According to doctors’ scheme
Mold testing Deterministic 16 min Opening hours linacs (all working days)
Linac sessions Deterministic 16 min Opening hours linacs (all working days)
One stop shop Deterministic 60 min Amsterdam: 60 min on working days, Flevo: 60 min on Friday
Table 3
Patient types as input for the simulation model.
Patient type Percentage Deviation from standard care path
Dermatology normal 1.1
Digestive normal 7.6
Digestive with PET-CT 2.6 PET-CT before CT scan
Digestive with beads 3.7 Beads, wait three days before CT scan
Gynecology with brachy 3.3 Tune sessions with brachytherapy
Gynecology with PET-CT 1.2 PET-CT before CT scan
Hematology normal 2.4
Lung normal 0.4
Lung with PET-CT and 4D-CT 3.2 PET-CT, next day 4D-CT scan
Mamma after surgery 18.7
Mamma with mold 3.7 Mold check before start sessions
Urology normal 2.8
Urology with gold beads 13.4 Gold beads, wait three days, CT scan
Urology with cystoscopy 1.7 Cystoscopy, wait one day, CT scan
Urology adjuvant 0.7 Tune sessions with start adjuvant therapy
Curative others 3.5
Palliative bone metastasis 9.6
Palliative others 20.3
Table 4
Patient characteristics as input for the simulation model.
Patient characteristic For which patients Description Percentage
Location All patients Amsterdam 73
Flevo 27
Consultation type Palliative others in Amsterdam Standard care path 49
One stop shop 51
Palliative bone metastasis in Amsterdam Standard care path 16
One stop shop 84
Palliative others in Flevo Standard care path 22
One stop shop 78
Palliative bone metastasis in Flevo Standard care path 38
One stop shop 62
Table 5
Results of the experiments, in average number of calendar days.
Experiment Regular patients Subacute patients
Lower bound of access time Access time (simulation) Lower bound of access time Access time (simulation)
Original situation 15.2 26.7 9.8 6.9
Exp. 1: optimal scheme 13.9 22.8 7.1 5.8
Exp. 2: breast/urology by all doctors 13.9 21.8 7.1 5.8
Exp. 3: CT scan earlier 12.7 21.5 5.7 5.0
Exp. 4: no contouring time slots 13.8 20.8 7.1 4.7
Exp. 5: additional capacity 15.2 25.2 9.8 6.1
Exp. 6: combination of exp. 2–4 13.9 18.6 7.1 4.3
impact of extending the capacity of the CT simulators and linacs (experiment 5). For experiments 2 and 4 we find that the simulated access times for regular patients are reduced with respect to
experiment 1, although the used doctors’ scheme is the same. This is due to the introduced flexibility of choosing between multiple doctors and the freedom to contour scans on any day of the week.
Fig. 4. The original doctors’ scheme and the schemes of experiments 1–3.
Fig. 5shows the percentages of access times that are within the
specified standards. Recall that the national standards prescribe that 80% of the subacute patients should be treated within seven days and 100% within ten days, and for regular patients that 80% should be treated within 21 days and 100% within 28 days. Gener-ally, we observe that all investigated interventions improve these percentages. For the regular patients, the percentages increase more than for subacute patients. This is mostly due to the fact that subacute patients can be treated by all doctors, such that the opti-mized doctors’ schemes do have less effect for these patients. The combination of interventions (experiment 6) is the most successful experiment with respect to meeting the standards, both for regu-lar and subacute patients, but additional interventions seem to be required to meet the access time standards.
Some of the experiments are expected to have a different effect on different patient types. A more detailed insight of the effects on the access times per patient type is presented inFig. 6for the original situation and experiments 1 and 2. The values in the left figure indicate the access times met by 80% of the patients, and the lines indicate the access time standards. Remaining differences are mostly due to additional stages that are required for some patient types.
We have also investigated the separate contribution of the different stages to the access times. It appears that the relatively largest reduction in access times is in the entry time (time between referral and consultation). The 80-percentiles of these entry times are displayed in the right part ofFig. 6. Observe that entry times are far more balanced over the patient types in experiment 1 and 2 than in the original situation. This indicates that the distribution of consultation slots over the week is an important factor in access time reduction.
6. Conclusion and discussion
In this paper, we have presented a methodology for optimizing capacity allocation of doctors’ activities in the radiotherapy care
process, such that compliance to the access time standards can be increased for all patients. Doctors carry out consultations and scan contouring according to a cyclic weekly scheme. In this study, alignment of these doctor’s schemes is done by matching daily consultation time slots with daily demand, spreading consultation time slots for each patient type over the week, aligning the time division of doctors over different activities, and taking the capacity allocation in consecutive stages into account.
The potential effectiveness of this methodology is demon-strated by its application to a case study in the Radiotherapy department of the AMC. Several interventions have been investi-gated. We have seen that all investigated interventions that include an optimized doctors’ scheme have a beneficial effect on the access times: access times are reduced considerably and differences in ac-cess times among different patient types are reduced. We have ob-served in the case study that introducing a new doctors’ scheme has a larger effect than increasing machine capacity. The results demonstrate that adequate capacity allocation can reduce access times for radiotherapy. Application of our model for improving the capacity allocation in radiotherapy care is very promising. The management of the Radiotherapy department in the AMC is well satisfied with the doctors’ schemes generated by the model. The resulting doctor’s schemes are currently being evaluated for im-plementation in the AMC.
Given the results of the AMC case, we are convinced that the application of our method can be valuable to many radiotherapy facilities. The AMC case is representative for hospitals in the Netherlands regarding facility size and care process characteris-tics. The case mix might be hospital-specific, with the consequence that in other hospitals the patient types each doctor can treat might be different. If doctors could treat more patient types, higher flex-ibility in assigning patients to doctors is expected, and therefore smaller differences in access times between patient types. The ILP approach is suitable for changing or adding constraints, and mod-ifying the objective function. Therefore, our model can readily be
Fig. 5. Evaluation of the access times with respect to the standards for subacute and regular patients.
Fig. 6. Access time and entry time per patient type, in the original situation and experiments 1–2.
customized to incorporate the particular characteristics and pref-erences of other radiotherapy care facilities.
Besides the logistical intervention we considered in this paper, further improvement is possible in the compliance to the access times standards of the radiotherapy care process. One of the suggestions for future research is to investigate priority rules or capacity reservation for certain groups of patients, for example (sub)acute patients. Also, a dynamic capacity allocation could be considered to control access times in periods with unusual demand or supply conditions. Furthermore, in the paper we have not taken into account that treatments have to be tuned with adjuvant therapies, such as chemotherapy and surgery. To obtain an integral view of cancer care, it would be valuable and advantageous to take this combination into account as well.
Competition over resources is an important issue in hospitals, as many care processes deal with aligning capacity of multiple disciplines and resources in interrelated consecutive care stages as well as with time division of resources over activities. Therefore, many relevant possibilities exist for our method to be applied in other applications than radiotherapy, for example in rehabilitation or rapid cancer diagnostics.
Acknowledgments
The authors are grateful to the personnel of the Radiotherapy department of the Academic Medical Center (AMC) Amsterdam for their involvement in the development and implementation of the method, and to Nikoletta Pittokopiti for her work on the simulation model. This research has been supported by the Twente Graduate School Fund and the Academic Medical Center.
Appendix. Mathematical formulation ILP model
This appendix describes, the mathematical formulation of the model introduced in Section4.1. The sets and indices are given
inTable 6. The parameters and variables used in the model are
displayed inTable 7. Before defining the constraints, we specify some stages that the model always contains inTable 8.
Table 6
Sets and indices in the ILP model.
Set Index Description
A a Doctors
P p Patient types
L l,l′
Treatment locations
S s,t Stages
B b Time slots per day
U u,u′ Time slots in the planning horizon
D d,d′
Time slots per week(=7|B|)
R r Referral days per week
Constraints
In the following constraints, the word ‘patient’ is used to refer to a patient of type p, referred on day r on location l.
Basic constraints
Doctor a can do at most one activity on time slot d, and only if he/she is available:
Xadl
+
Yadl≤
cadl∀
a,
d,
l.
(1)All patients are administered in the system in the last time slot of their referral day r:
Wprlsu
=
1∀
p,
r,
l,
s=
1,
u=
r|
B|
.
(2)Each stage s has to take place in at most one time slot for each patient:
u
Wprlsu
≤
1∀
p,
r,
l,
s.
(3)Constraints on the scheme construction
A doctor has a prescribed number of consultation time slots and contouring time slots per week. The number of consultation time slots of doctor a on location l is a given parameter:
d
Xadl
=
gal∀
a,
l.
(4)The total number of contouring time slots of doctor a is a given parameter:
d,l
Table 7
Parameters and variables in the ILP.
Notation Description
Binary parameters
cadl 1 if doctor a is available at time slot d on location l, 0 otherwise dpal 1 if patient type p can be treated by doctor a on location l, 0 otherwise fsdl 1 if stage s is available/opened at time slot d on location l, 0 otherwise
vpst 1 if patient type p has to visit stage t after stage s, 0 otherwise
wp 1 if patient type p is regular, 0 if patient type p is subacute Integer parameters
qpstl Prescribed slots for type p between stages s and t on location l (0 if consecutive slots)
gal Number of consultations for doctor a per week on location l
ha Number of contouring time slots for doctor a per week (on either location)
ml Maximum number of simultaneous consultation time slots at location l
kl Maximum number of simultaneous contouring time slots at location l
Real parameters
nprl Average number of arriving patients of type p at location l at referral day r bp Weight factor for ‘importance’ of short lower bound for patient type p
α, β Weight factors for the objective function
Binary variables
Xadl 1 if doctor a is scheduled for consultations on time slot d on location l, 0 otherwise Yadl 1 if doctor a is scheduled for contouring on time slot d on location l, 0 otherwise Zprla 1 if patient type p arriving at day r on location l is assigned to doctor a, 0 otherwise Wprlsu 1 if patient type p arriving at day r on location l is assigned stage s at time slot u, 0 otherwise Cprlad 1 if patient p,r,l could have consultation at time slot d if assigned to doctor a, 0 otherwise Dprlad 1 if patient p,r,l could have contouring at time slot d if assigned to doctor a, 0 otherwise Integer variables
ATprl Access time of patient type p arriving at day r on location l
Gplu Number of consultation time slots for patient type p on location l on day u Real variable
Fr Difference between average arrival rate at day r and no. of consultation time slots at day r+1
Table 8
Standard stages in the ILP model.
s Stage
1 Referral
2 Consultation
|S| −1 Scan contouring
|S| First session
The number of consultation time slots at any time slot d on location
l is restricted to the facility’s capacity:
a
Xadl
≤
ml∀
d,
l.
(6)The number of doctors contouring at any time slot d on location l is restricted to the facility’s capacity:
a
Yadl
≤
kl∀
d,
l.
(7)Constraints for patient–doctor allocation
A patient is assigned to exactly one doctor a:
a
Zprla
=
1∀
p,
r,
l.
(8)A patient can be assigned to doctor a, only if the patient type can be treated by the doctor on this location:
Zprla
(
1−
dpal) =
0∀
p,
r,
l,
a.
(9)A patient should be assigned to the first available doctor special-ized in the patient type:
0
≥
u−1
d′=r|B|+1 Gpld′− |
P| |
L| |
U|
(
1−
Wprlsu)
∀
p,
r,
l,
u,
s|
s=
2,
u>
r|
B| +
2.
(10)Constraints for patient-appointment allocation
A patient can have a consultation at time slot d, only if he/she is assigned to doctor a that has consultations during that time slot
and that location:
a
Cprlad
≥
Wprlsu∀
p,
r,
l,
u,
s=
2,
d=
(
u−
1)
mod(
7|
B|
) +
1.
(11) A patient can have contouring at time slot d, only if he/she is as-signed to doctor a that has a contouring task during that time slot:
a
Dprlad
≥
Wprlsu∀
p,
r,
l,
u,
s= |
S| −
1,
d=
(
u−
1)
mod(
7|
B|
) +
1.
(12) Stage s can only take place at location l on time slot u, if the stage is available on that time slot at that location (for all stages except referral, consultation and contouring):
p,r
Wprlsu
(
1−
fsdl) =
0∀
l,
s>
2 and s̸= |
S| −
1,
d=
(
u−
1)
mod(
7|
B|
) +
1.
(13) If a stage is not necessary for patient type p, then it should not take place:
u,l,r Wprlsu≥
tv
pts|
L| |
R| |
U|
∀
p,
s>
1.
(14)Stages have to be fulfilled in the right order and with the right time constraints between the stages for each patient type p. (If stage t has to take place after s (that is,
v
pst=
1), then the number of timeslots between stage s and t must be more than or equal to qpstl
+
1.)
u
(
u(
Wprltu−
Wprlsu)) − (
1+
qpstl)
+
(
1−
v
pst) |
U| ≥
0∀
p,
r,
l,
s,
t.
(15) For regular patients, treatment cannot start on Friday:w
pWprlsu=
0∀
r,
l,
s= |
S|
,
(16)(
u−
1)
mod(
7|
B|
) +
1>
4|
B| +
1 anda Frl
=
a,d∗ Xadl
p,r nprl
a gal
−
p nprl,
(19)where d∗
=
(
2r)
Mod(
5|
B|
) +
1 to(
2r+
1)
Mod(
5|
B|
) +
1.
The variables Cprladand Dprladare defined as follows:Cprlad
≤
Zprla∀
p,
r,
l,
d,
a,
(20)Cprlad
≤
Xadl∀
p,
r,
l,
d,
a,
(21)Cprlad
≥
Zprla+
Xadl−
1∀
p,
r,
l,
d,
a,
(22)Dprlad
≤
Zprla∀
p,
r,
l,
d,
a,
(23) Dprlad≤
l′ Yadl′∀
p,
r,
l,
d,
a,
(24) Dprlad≥
Zprla+
l′ Yadl′−
1∀
p,
r,
l,
d,
a.
(25) Objective functionThe objectives are to minimize the total weighted lower bound of the access times for all patients p
,
r,
l (with weight factorα
) and to minimize the difference between daily supply of the consulta-tion time slots and daily patient arrivals (with weight factorβ
):min
α
p(
bp
r,l nprlATprl) + β
r Frl.
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